2513. Minimize the Maximum of Two Arrays

Problem Description

The essence of this problem is to fill two initially empty arrays arr1 and arr2 with distinct positive integers that are not divisible by respective divisors: divisor1 for arr1 and divisor2 for arr2. Additionally, we must ensure that no integer is present in both arrays. The goal is to figure out the minimum value of the maximum integer that can be present in either of the arrays while meeting the following conditions:

• arr1 must contain uniqueCnt1 distinct positive integers, none of which should be divisible by divisor1.
• arr2 must contain uniqueCnt2 distinct positive integers, none of which should be divisible by divisor2.
• There must not be any overlap between the integers in arr1 and arr2.

The challenge thus revolves around selecting numbers carefully to minimize the highest number placed in either array, all the while respecting the given divisibility rules.

Intuition

The solution to this problem can be derived using a binary search on the range of possible maximum values for the arrays. Since a binary search requires a sorted range and certain conditions to find a target, we define the conditions based on understanding how many numbers will be allowed in each array up to a certain value x. The binary search begins with an initial range from 0 to an arbitrarily large number (in this case, 10^10), looking for the smallest x that satisfies our conditions.

Firstly, we need a function that checks if x can be the maximum possible integer in either array. I will define f(x), which calculates the count of distinct non-divisible integers up to x for arr1 and arr2 separately, and a combined count for both arrays together. It does so by incorporating the floor division and modulo operations for each divisor as well as the lowest common multiple (LCM) of both divisors.

The intuition behind these calculations is that every divisor1 consecutive numbers, divisor1 - 1 will be admissible for arr1 since one will be divisible and thus excluded. The same logic applies to arr2 with divisor2. For common elements to be excluded from both arrays, we consider the LCM of both divisors, as any number divisible by the LCM would be divisible by both divisor1 and divisor2.

The function f(x) checks whether, up to the number x, there are at least uniqueCnt1 non-divisible numbers for arr1, uniqueCnt2 for arr2, and a combined uniqueCnt1 + uniqueCnt2 for both with overlap removed. If this condition holds, then x is a candidate for the minimum possible maximum integer.

The use of bisect_left in Python is how we do our binary search here. It essentially searches for the place where True would be inserted in the range to keep it sorted, as per the f(x) condition. This gives us the smallest value x that satisfies f(x).

By initializing the binary search with a large enough range, we ensure that we can find a cut-off point where f(x) transitions from False to True, indicating that we've found the minimum possible maximum integer for the arrays.

Learn more about Math and Binary Search patterns.

Solution Approach

The implementation of the solution utilizes several important concepts from algorithms and mathematics:

1. Least Common Multiple (LCM): Before we start the binary search, we need to find the least common multiple of the two divisors. This is crucial as we need to exclude numbers divisible by both divisor1 and divisor2 when counting the numbers admissible in both arrays combined. As the LCM is not provided in the solution code, it must be implemented in helper functions or use external libraries.

2. Counting Function (f): We define a counting function f(x) that uses arithmetic to determine the count of distinct non-divisible integers up to a certain value x for each array, as well as combined. In more detail:

   • We calculate cnt1 by determining how many groups of divisor1 can fit into x (given by x // divisor1) and then adding the remaining numbers (given by x % divisor1). We exclude one number for each full group since one number will be divisible by divisor1.
   • Similarly, cnt2 is calculated for arr2 and divisor2.
   • For the combined count cnt, we apply the same logic using the LCM of the two divisors since we want to exclude numbers that are disallowed in both arrays.

3. Binary Search Algorithm: With bisect_left, we are efficiently conducting a binary search. The range [0, 10**10] is just a representation of an upper bound that we are confident is much higher than our target maximum integer. Binary search is applied by using f(x) as the key function for bisect_left which allows us to find the smallest number x for which f(x) returns True, i.e., the counting function indicates that both arr1 and arr2 have their required distinct positive integers.

4. Binary Search with a Custom Condition (f): Instead of looking for an actual value in a sorted list, as is typical with binary search, we are using the binary search to find the point at which a condition changes from False to True. This binary search is abstract in the sense that it operates on the truth value of f(x) over a range of numbers.

By combining the above techniques, the solution finds the minimum possible maximum integer with the least amount of computation necessary, avoiding the need to directly generate the arrays or to iterate over large ranges of numbers, which would be inefficient.

The most crucial part of the solution is ensuring that the key function f(x) accurately reflects the conditions we need for arr1 and arr2. Once f(x) is correctly defined, bisect_left takes over to execute the binary search, taking advantage of the fact that f(x) will be False until it isn't—at which point we've found our minimum maximum.

It's worth noting that mathematical intuition is key in converting the problem into one that can be solved with binary search. Specifically, understanding how to calculate the number of admissible numbers given the constraints on divisibility and ensuring the arrays remain disjoint are the fundamental subproblems addressed by the counting function f(x).

Example Walkthrough

Let's consider an example scenario to illustrate the solution approach:

Assume we have divisor1 = 2, divisor2 = 3, uniqueCnt1 = 2, and uniqueCnt2 = 2.

We need to create two arrays, arr1 and arr2, such that:

• arr1 contains two distinct integers not divisible by 2.
• arr2 contains two distinct integers not divisible by 3.
• None of the integers are overlapping between arr1 and arr2.

Step-by-Step Process

1. Finding the Least Common Multiple (LCM):

   • The LCM of 2 and 3 is 6. We need this for counting common numbers that would be excluded from both arrays.

2. Defining the Counting Function (f):

   • Let's say we are checking if x = 7 can be the maximum integer in either array.
   • For divisor1 = 2, up to x = 7, the numbers 1, 3, 5, 7 (four numbers) are not divisible by 2.
   • For divisor2 = 3, up to x = 7, the numbers 1, 2, 4, 5, 7 (five numbers) are not divisible by 3.
   • For numbers up to x = 7 that are not divisible by 2 or 3 (using LCM of 6), we get 1, 5, 7 (three numbers).

3. Applying Binary Search with Custom Condition (f):

   • Since we require at least 2 numbers for each array and to avoid overlap, we need to confirm if the combined total without overlap is at least 4. For x = 7, we have 3 such numbers, so f(7) is False.
   • Using binary search, we increase x and check again. Let's say x = 8. We repeat the counting:
     • For divisor1 = 2, the numbers are 1, 3, 5, 7 (still four numbers).
     • For divisor2 = 3, the numbers are 1, 2, 4, 5, 7, 8 (six numbers).
     • Using the LCM of 6, the numbers are 1, 5, 7 (still three numbers).

4. Finding the Transition Point:

   • For x = 8, f(8) is False as we still don't have at least 4 non-overlapping numbers. We proceed with the binary search.
   • Increment x and check again. At x = 10, we repeat the counting:
     • Non-divisible by 2: 1, 3, 5, 7, 9 (five numbers).
     • Non-divisible by 3: 1, 2, 4, 5, 7, 8, 10 (seven numbers).
     • Non-divisible by both (LCM = 6): 1, 5, 7, 11 (four numbers – 11 is now included since x = 10).
   • Since f(10) is True for the first time (we have at least 2 numbers for each array without overlap), the binary search concludes, and we consider 10 to be the smallest possible maximum integer that satisfies all conditions.

Therefore, for our example with the chosen parameters, the minimum possible maximum integer that can be placed in either array while satisfying all conditions is 10. This example demonstrates the method through which we can utilize a binary search to efficiently solve for the required parameters.

Solution Implementation

Time and Space Complexity

The given Python code aims to find the minimum number x that satisfies certain divisibility conditions. The code is expected to use the bisect_left function from the bisect module to perform a binary search for the smallest value of x where function f(x) returns True. Unfortunately, the provided code contains errors and it's incomplete (there's a reference to an undefined variable divisor inside the f(x) function). Assuming these issues were fixed and the lcm function (least common multiple) and bisect_left were correctly implemented, we can proceed with the time and space complexity analysis.

Time complexity:

1. bisect_left: The binary search implemented by bisect_left has a time complexity of O(log N), where N is the size of the range. In this case, N is set to 10**10, which is a constant, so the bisect_left call alone would have a complexity of O(log 10**10) which simplifies to O(log 10) due to the base not being specified in the Big O notation.

2. Function f: This function is called by bisect_left at each step of the binary search. The operations inside function f are performed in constant time (basic arithmetic operations) and do not depend on the size of the input.

Combining the two above, we get O(log 10) multiplied by the time complexity of f, which is O(1), resulting in O(log 10). However, the improper references in the f(x) function can alter this analysis if, for example, computing the lcm has a non-constant complexity.

Space complexity:

1. bisect_left: The space complexity of the binary search is O(1) since it doesn't allocate any additional space that grows with the input.

2. LCM computation: If the lcm is computed each time f(x) is called, and the computation of lcm is not in O(1) space, this would affect the space complexity. If the lcm is computed once and used multiple times with a constant space complexity, the space complexity will remain O(1).

3. Function f: Since the function f only uses a fixed number of variables and does not use any data structures that scale with the size of the input, its space complexity is O(1).

Considering all the above, the overall space complexity of the code is O(1), provided the computationally-intensive tasks such as finding the least common multiple have constant space complexity and are computed outside of the f(x) function or are otherwise cached.

It's important to note that the presence of the range(10**10) in the bisect_left function does not impact space complexity since range in Python 3 creates a range object that does not generate all numbers in memory but computes them on-the-fly.

Learn more about how to find time and space complexity quickly using problem constraints.